Chapter 17 is devoted to an analysis of Maxwellian electrodynamics (Sect. 17.1) in arbitrary Einsteinian spacetime manifolds \(\mathcal {M}\) in terms of the Maxwell 2-form \(F\in \Gamma \Lambda ^{2}\mathcal {M}\) , the Maxwell induction 2-form \(G\in \Gamma \Lambda ^{2}\mathcal {M}\) , the Maxwell polarisation 2-form \(\Pi \in \Gamma \Lambda ^{2}\mathcal {M}\) , the electromagnetic electric current 3-form \(\mathcal {J}\in \Gamma \Lambda ^{3}\mathcal {M}\) and an arbitrary observer Frame \(Z\in \Gamma T\mathcal {M}\) . This data enables one to split the Maxwell field equations with sources in an arbitrary spacetime into equations involving instantaneously observable electric and magnetic spacetime fields relative to Z. Section 17.1 concludes with a formulation of a (Lorentz) force law for the motion in vacuo of a charged particle in a background electromagnetic field F. Section 17.2 introduces the notion of a spacetime conformal Killing vector field and demonstrates that in the absence of media polarisation (i.e. \(G=\epsilon _{0}F\) ) the Maxwell field equations are “conformally covariant”. In Sect. 17.3, the notion of local \(\text{U}(1)\) electromagnetic gauge covariance is introduced for fields on simply-connected spacetime domains where one can find a class of 1-forms \(A\in \Gamma \Lambda ^{1}\mathcal {M}\) such that \(F=dA\) . The existence of the class \(\{A\}\) leads to the definition of \(\text{U}(1)\) gauge covariant derivatives of elements in the class of complex scalar fields that can be used to construct \(\text{U}(1)\) gauge invariant expressions for electrically charged, interacting field systems. In Sect. 17.4, the notion of the local \(\text{U}(1)\) gauge covariant derivative is generalised to the notion of a local \(\text{GL}(n,\mathbb {R})\) gauge covariant derivative. Unlike the group \(\text{U}(1)\) , the group \(\text{GL}(n,\mathbb {R})\) of real-valued, linear transformations that transform one n-dimensional local basis of vector fields to another n-dimensional local basis of vector fields is non-abelian. In Sect. 17.4, we construct local \(\text{GL}(n,\mathbb {R})\) gauge covariant indexed antisymmetric canonically ordered tensors with an arbitrary finite number of upper and lower indices, associated with an arbitrary connection \(\nabla \) . In particular, this enables one to express the first and second Bianchi identities (Sect. 6.6 ) more compactly and streamlines the notation for discussing various generalised theories of non-Einsteinian gravitation.

Split Structures for Electromagnetic Fields and Gauge Covariances

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发表日期：

2025年

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Robin W. Tucker,
Timothy J. Walton

摘要

Chapter 17 is devoted to an analysis of Maxwellian electrodynamics (Sect. 17.1) in arbitrary Einsteinian spacetime manifolds \(\mathcal {M}\) in terms of the Maxwell 2-form \(F\in \Gamma \Lambda ^{2}\mathcal {M}\) , the Maxwell induction 2-form \(G\in \Gamma \Lambda ^{2}\mathcal {M}\) , the Maxwell polarisation 2-form \(\Pi \in \Gamma \Lambda ^{2}\mathcal {M}\) , the electromagnetic electric current 3-form \(\mathcal {J}\in \Gamma \Lambda ^{3}\mathcal {M}\) and an arbitrary observer Frame \(Z\in \Gamma T\mathcal {M}\) . This data enables one to split the Maxwell field equations with sources in an arbitrary spacetime into equations involving instantaneously observable electric and magnetic spacetime fields relative to Z. Section 17.1 concludes with a formulation of a (Lorentz) force law for the motion in vacuo of a charged particle in a background electromagnetic field F. Section 17.2 introduces the notion of a spacetime conformal Killing vector field and demonstrates that in the absence of media polarisation (i.e. \(G=\epsilon _{0}F\) ) the Maxwell field equations are “conformally covariant”. In Sect. 17.3, the notion of local \(\text{U}(1)\) electromagnetic gauge covariance is introduced for fields on simply-connected spacetime domains where one can find a class of 1-forms \(A\in \Gamma \Lambda ^{1}\mathcal {M}\) such that \(F=dA\) . The existence of the class \(\{A\}\) leads to the definition of \(\text{U}(1)\) gauge covariant derivatives of elements in the class of complex scalar fields that can be used to construct \(\text{U}(1)\) gauge invariant expressions for electrically charged, interacting field systems. In Sect. 17.4, the notion of the local \(\text{U}(1)\) gauge covariant derivative is generalised to the notion of a local \(\text{GL}(n,\mathbb {R})\) gauge covariant derivative. Unlike the group \(\text{U}(1)\) , the group \(\text{GL}(n,\mathbb {R})\) of real-valued, linear transformations that transform one n-dimensional local basis of vector fields to another n-dimensional local basis of vector fields is non-abelian. In Sect. 17.4, we construct local \(\text{GL}(n,\mathbb {R})\) gauge covariant indexed antisymmetric canonically ordered tensors with an arbitrary finite number of upper and lower indices, associated with an arbitrary connection \(\nabla \) . In particular, this enables one to express the first and second Bianchi identities (Sect. 6.6 ) more compactly and streamlines the notation for discussing various generalised theories of non-Einsteinian gravitation.